CURRENT ELECTRICITY — Physics STD 12 Science — Question
Bihar BoardEnglish MediumSTD 12 SciencePhysicsCURRENT ELECTRICITY2 Marks
Question
By explaining the (Electric) current density, derive Ohm's law in vector form.
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Answer
→ Current density : "The electric current flowing per unit cross sectional area perpendicular to the current is called the (electric) current density $(\vec{j}) . "$ → Suppose current I is flowing through the cross sectional area A , then electric current density $\therefore j=\frac{ I }{ A }$ → The SI unit of current density is $\frac{ A }{ m ^2}$. → It is a vector quantity and the dimensional formula is $M ^0 L^{-2} T^0 A^1$. → If the magnitude of uniform electric field in the conductor of length $l$ is E , then the potential difference across its ends, $V = E l$ → According to the ohm's law $\begin{array}{rlrl} V & = IR \\ \text { but } & R & =\frac{ ρl}{A} \\ & V & =\frac{ Iρl}{A} \\ \therefore & E l & =\frac{ Iρl}{A} \\ \therefore & E & =\frac{ Iρ }{ A } & (\because V = E l) \end{array}$ → Using equation (1), $\therefore E =j ρ$ → The reciprocal of resistivity is called the conductivity of material. $\therefore \sigma=\frac{1}{ρ}$ → where $\sigma=$ conductivity of material from equation (2) and (3), $\begin{array}{l} \therefore E =\frac{j}{\sigma} \\ \therefore j=\sigma E \end{array}$ → This equation can be written in vector form $\therefore \vec{j}=\sigma \overrightarrow{ E }$ → This equation is called vector form of Ohm's law.
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